GMAT Problem Solving—Be Flexible in Your Approach (and Know What You Need to Know)!

Anytime you have roots in the denominator, an important best practice in algebra should be applied: remove the root by multiplying by one, a process called rationalizing the denominator. As an example, if you have , then you would simply multiply by one as follows to simplify the expression and remove the root from the denominator:

This process is made more difficult in this official question with two mechanisms:

1. 1. • The denominator does not just contain individual terms with roots but also integers or multiple roots added together. This makes for a more difficult version of rationalizing the denominator in which you must recognize the difference of squares and” multiply by one” using the conjugate of the denominator. Even though this is harder, these are both core best practices that you learned in algebra in high school and that you must know for the GMAT. This skill has been tested so many times on the GMAT that you should recognize two things immediately: you should use the conjugate and the denominators will simply disappear with the numbers they have used.
      • Three terms with roots in the denominator are being added together, so the test-maker entices students to try to find a common denominator or take some other incorrect approach.

Consider the first term of the three being added together in this problem: . When you factor the difference of squares into its conjugates, you get . With this knowledge (and some reverse-engineering) you can see that if you multiply by one in the form then all the roots will disappear in the denominator (and in this often-used case on the GMAT, the entire denominator!) as shown here:

The important point with a manipulation like this is that you simply must recognize what to do! We cover these types of important math skills in detail in our Refresh Modules and then you need to practice them with questions like this. Once you see what to do on the first term, then just do the same type of manipulation on each fraction individually and add the simplified terms together.

The 2nd term being added together is simplified in the same way as follows:

And the 3rd term:

With each of the three fractions simplified and the denominators disappearing, you are simply adding together the following three terms:

The correct Answer is thus (E).

While these three steps look tedious on paper, the reality is that a lot of people taking the GMAT are going immediately to the last step shown above without any written work. You want to be one of those people!

If you don’t know what to do on this problem algebraically (and the point of this example is that you should!), it is important to note that this question can also be solved cleverly using answer choices and simply estimating the roots. Since the first four answer choices are all less than 1/2, you know the answer must be (E). Estimating the two roots in the question stem allows you to see that the sum of the three expressions will get close to 1, and none of the other answers are close. If you solved this question with this technique, good for you! However, this question could easily contain 6/7 as an answer, and then you would be in trouble.

As you prepare for the exam, pay special attention to any misses on questions like this that just require math knowledge. They are easier to prepare for and it is important that you get them right. With difficult abstract problems involving lots of red herrings or tricky wording, you simply can’t get win them all, but for these types of questions, you can develop complete mastery.

This example is a classic type 2 question—it feels abstract and you must read carefully. You can be comfortable with most percent questions on the GMAT and still get this wrong (or waste a lot of time) if you don’t choose the right approach.

If you search the internet for explanations on this question, you see everyone explaining one tedious algebra step after another AND you see many people who have either botched that algebra or made a mistake setting up the percent change. In 20 years of preparing people for the test, I have only seen a few variable-in-answer choice percent questions for which algebra was a better approach than number picking. Here the algebra is not as tedious as in other questions of this kind, but number picking is unquestionably easier.

As a best practice for the exam, always take a little time to decide on your approach (algebra, conceptual thinking, backsolving, or number picking) before jumping into a question. Don’t swim upstream with a long math approach when you can take advantage of answers or use your own numbers. As we teach in our curriculum, whenever you see percent change questions with variables, try number picking first and only go to algebra if that is not working. When number picking, it is important that you are careful with the number(s) that you choose for variables—that is, anticipate and use numbers that will make solving the question as easy as possible. On harder number picking questions, you may choose the wrong numbers first and only realize which ones will work better as you move into the question.

In this example, you need to imagine a number of hours for x and then calculate the percent change with that value. Remember that with % change questions, you are always putting the difference over the original number, so you would be smart to choose a value for x that would make the original number a nice round number. Selecting x = 5 would do just that: if one year ago, the window washing service worked for 5 hours, then the fee was $100 + (5 x $30) or $250. The new fee for 5 hours would be $20 + 5 x $50 or $270. Then you just need to calculate the % change from 250 to 270. Take the difference of 20 and compare to the original of 250 to see: which equals or an increase of 8%.

The final step after you solve with the number(s) you have chosen is to plug that number into each answer, looking for your solution, in this case 8%. By plugging 5 into each answer, it is clear that (D) and (E) are wrong as they would be negative. (A) is way too big and (C) would leave 27 in the denominator (i.e. not reduce to 8) so the correct answer must be (B).

Thinking about this question broadly, it is really quite simple with number picking as long as you pick a good number!!!! One risk in number picking is that you get buried in awkward calculations. Imagine if you picked say 3 or 7 for x. With 3, you would be starting at $190 and calculating the % change to $170. Ugly. With 7 it would be $310 to $370. Also, ugly.

Number picking is an essential strategy for GMAT word problems with variables in answers and for many other question types (percent questions or others in which the starting number can be anything). You must practice and hone this strategy in the same way that you do with certain quant skills and calculations, but most people are not doing that in their preparation. As you move through official questions, take the time to consider alternative approaches after you have solved a question, particularly if your method seemed tedious or time-consuming. As a final exercise, think about how easy you can make a problem like this compared to how it first seems:  if I asked you what the percent change was from 250 to 270, I am confident that all of you could get it correct in less than a minute!